It is also not hard to see that there is a subsetSSS of AAA such that SSS is affinely independent and the span of SSS is AAA. If AAA is finite dimensional, then SSS is finite, and that every element of AAA can then be expressed uniquely as a finite affine combination of elements of SSS. Because of the existence and uniquess of this expression, we can write every element v∈AvAv\in A as

The expression (k1,…,kn)subscriptk1normal-…subscriptkn(k_{1},\ldots,k_{n}) is called the barycentric coordinates of vvv (given SSS). Each kisubscriptkik_{i} is called a component of the barycentric coordinates of vvv.

Remarks.

Unlike the Euclidean space, v+wvwv+w and k⁢vkvkv with 1≠k∈F1kF1\neq k\in F defined by coordinate-wise addition and scalarmultiplication do not make sense in an affine space. If v=(k1,…,kn)vsubscriptk1normal-…subscriptknv=(k_{1},\ldots,k_{n}) and w=(m1,…,mn)wsubscriptm1normal-…subscriptmnw=(m_{1},\ldots,m_{n}), then (k1+m1)+⋯+(kn+mn)=2subscriptk1subscriptm1normal-⋯subscriptknsubscriptmn2(k_{1}+m_{1})+\cdots+(k_{n}+m_{n})=2 and v+w:=(k1+m1,…,kn+mn)assignvwsubscriptk1subscriptm1normal-…subscriptknsubscriptmnv+w:=(k_{1}+m_{1},\ldots,k_{n}+m_{n}) would be meaningless.

Similarly, ?:=(0,…,0)assign00normal-…0\boldsymbol{0}:=(0,\ldots,0) does not exist in an affine space for the simple reason that ?0\boldsymbol{0} is not an affine combination of any subset of an affine space AAA (0≠1010\neq 1). The notion of an origin has no place in an affine space.

However, any finite affine combination of any set of points in an affine space is always a point in the space. This can be easily illustrated by the use of barycentric coordinates. For example, take v=(k1,…,kn)vsubscriptk1normal-…subscriptknv=(k_{1},\ldots,k_{n}) and w=(m1,…,mn)wsubscriptm1normal-…subscriptmnw=(m_{1},\ldots,m_{n}). Let

If FFF is ordered, then we can form sets in an affine space consisting of points with non-negative barycentric coordinates. Given a set SSS of affinely independent points, a set GGG is called the affine polytope spanned by SSS if GGG consists of all points that are in the span of SSS and have non-negative barycentric coordinates via SSS. A point in GGG is said to lie on the surface of the polytope if it has at least one zero component, otherwise it is an interior point (having all positive components). In the language of algebraic topology, this is also known as a simplex.